2nd Reading

c World Scientific Publishing Company DOI:10.1142/S0219887812500041

PSEUDOZ SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS

CARLO ALBERTO MANTICA

Physics Department, Universit`a degli Studi di Milano Via Celoria 16, 20133, Milano, Italy

carloalberto.mantica@libero.it

YOUNG JIN SUH Department of Mathematics Kyungpook National University

Taegu 702-701, Korea yjsuh@knu.ac.kr

Received 23 February 2011 Accepted 27 July 2011

Published

In this paper we introduce a new notion ofZ-tensor and a new kind of Riemannian man- ifold that generalize the concept of bothpseudoRicci symmetric manifold andpseudo projective Ricci symmetric manifold. Here theZ-tensor is a general notion of theEinstein gravitational tensorin General Relativity. Such a new class of manifolds withZ-tensor is namedpseudoZsymmetric manifold and denoted by (PZS)_n. Various properties of such ann-dimensional manifold are studied, especially focusing the cases withharmonic curvature tensors giving the conditions ofclosenessof the associated one-form. We study (PZS)_nmanifolds withharmonicconformal and quasi-conformal curvature tensor. We also show the closeness of the associated one-form when the (PZS)_nmanifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms,Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces,Bull. Soc. Math. Belg. Ser. A 44(1992) 1–34]). Finally, we study some properties of (PZS)₄spacetime manifolds.

Keywords: Pseudo Ricci symmetric manifolds; pseudo projective Ricci symmetric;

conformal curvature tensor; quasi-conformal curvature tensor; conformally symmetric;

conformally recurrent; Riemannian manifolds.

Mathematics Subject Classification 2010: 53C15, 53C25

1. Introduction

In 1988 Chaki [4] introduced and studied a type of non-flat Riemannian manifold whose Ricci tensor is not identically zero and satisfies the following equation:

∇kR_j= 2A_kR_j+A_jR_k+AR_kj. (1.1)

Such a manifold is calledpseudoRicci symmetric,A_k is a non-null covector called associated 1-form,∇is the operator of covariant differentiation with respect to the metricg_k and the manifold is denoted by (PRS)_n. Here we have defined the Ricci tensor to beR_k =−R_mk^m [30] and the scalar curvatureR=g^ijR_ij. This notion of pseudoRicci symmetric is different from that of Deszcz [14,15]. In [5] the authors considered conformally flatpseudoRicci symmetric manifolds, where the conformal curvature tensor

C_jk^m =R^m_jk+ 1

n−2(δ^m_jR_k−δ_k^mR_j+R_j^mg_k−R^m_k g_j)

− R

(n−1)(n−2)(δ_j^mg_k−δ^m_k g_j) (1.2) vanishes, that is, C_jk^m = 0. It may be scrutinized that the conformal curvature tensor vanishes identically forn= 3 [21]. In [2], a (PRS)_n withharmoniccurvature tensor, that is, ∇_mR^m_jk = 0 and with harmonic conformal curvature tensor, that is,∇mC_jk^m = 0 was studied (see [3]).

Such a notion of harmonic is related to the co-closeness of the curvature ten- sor. From this, together with the notion of closeness of the associated 1-form in (1.1), it gives us some important geometric meanings in the theory ofYang–Mill’s Connections,Harmonic MappingsandMathematical Physics, in particular, inEin- stein’s Relativity. From such a point of view, in this paper, we mainly consider the closenessof associated 1-forms for some generalized curvature tensors.

On the other hand, Suh, Kwon and Yang [27] introduced the notion of confor- mal curvature-like tensor on a semi-Riemannian manifold and have given a complete classification of conformally symmetric semi-Riemannian manifold with generalized non-null stress–energy tensor. More generally, Suh and Kwon [25] considered the notion of conformally recurrent semi-Riemannian manifolds with harmonic con- formal curvature tensor and gave another generalization of conformal symmetric Riemannian manifolds. Moreover, in [26] due to Suh, Kwon and Pyo the impor- tance of the closeness for the associated curvature-like 2-form corresponding to each concircular, projective and conformal curvature-like tensor defined on semi- Riemannian manifolds was remarked respectively.

Now let us consider a generalization of condition (1.1) introduced in a paper by Chaki and Saha [9]. They considered the so-called projective Ricci tensor P_k obtained by a suitable contraction of the projective curvature tensor P_jkm [16].

More precisely, one obtains

P_j = n n−1

R_j−R ng_j

, (1.3)

whereR=g^ijR_ij denotes the scalar curvature.

Obviouslyg^jP_j= 0. The generalization defined in [9] is thus written as

∇kP_j = 2A_kP_j+A_jP_k+A_lP_kj. (1.4)

This kind of manifold is called pseudo projective Ricci symmetric and denoted by (PPRS)_n. Recently in [7, 10] a further generalization of condition (1.1) was considered. More precisely a manifold whose Ricci tensor satisfies the condition

∇_kR_j = (A_k+B_k)R_j+A_jR_k+AR_kj (1.5) is defined. Such a manifold is called almostpseudo Ricci symmetric and denoted by (APRS)_n. HereA_k andB_k are non-null covectors.

In [10] the properties of conformally flat (APRS)_n are studied and the authors pointed out the importance of pseudo Ricci symmetric manifolds in the theory of General Relativity. It is therefore worthwhile to undertake the study of an n-dimensional manifold that generalizes both the (PRS)_n and the (PPRS)_n mani- folds. In this paper we define a generalized (0,2) symmetricZ-tensor given by

Z_k =R_k+φg_k, (1.6)

whereφdenotes an arbitrary scalar function. It is worth to notice that theZ-tensor allows us to reinterpret many well-known structures on Riemannian manifolds. In fact one can check that aZ flat Riemannian manifold is simply an Einstein space.

If a Z recurrent Riemannian manifold is considered i.e. a space satisfying the condition

∇_iZ_k=λ_iZ_k one can easily find that this condition is equivalent to

∇_iR_k=λ_iR_k+ (n−1)µ_ig_k

with the choice (n−1)µ_i =λ_iφ− ∇_iφ. So the manifold reduces to a generalized Ricci recurrent manifold [12] and if λ_iφ− ∇iφ = 0 a Ricci recurrent manifold is recovered (see also [19]).

If we consider a Riemannian manifold withZ-tensor of Codazzi type(see [13]), that is, with the property

∇kZ_j=∇jZ_k one can easily find that this condition is equivalent to

∇kR_j− ∇jR_k = (∇jφ)g_k−(∇kφ)g_j.

Transvecting the previous relation with g^j we get ∇k(R+ 2(n−1)φ) = 0 and finally

∇_kR_j− ∇_jR_k= 1

2(n−1)[(∇_kR)g_j−(∇_jR)g_k].

A manifold with Z-tensor of Codazzi type is thus a nearly conformal symmetric manifold (NCS)_n: this condition was introduced and studied by Roter [22] and Suh, Kwon and Yang [27]. Conversely, an (NCS)_n manifold has a Z-tensor ofCodazzi typeif the condition∇k(R+ 2(n−1)φ) = 0 is satisfied.

One can observe that the n-dimensional Einstein equations with cosmologi- cal constant Λ may be written in the same form Z_k = kT_k with the choice

φ=−^R₂ + Λ. This choice comes naturally from the condition ∇T_k = 0 for the stress–energy tensor and the second Bianchi identity that gives∇k(^R₂ +φ) = 0. In this case the vacuum solutionZ = 0 implies Λ =Rⁿ⁻²_2n and thus an Einstein space.

So the generalizedZ-tensor may be thought as ageneralized Einstein gravitational tensorwith arbitrary scalar functionφ.

Finally one can also notice that Einstein’s equations in General Relativity (n= 4) can be written in the suggestive form

Z_k=kT_k

with the choiceφ=−^R₂ withoutcosmological constant, that is, Λ = 0. In this case the tensor Z is said to be the Einstein gravitational tensor, T the stress–energy tensorandka certaingravitational constantof a spacetimeM, respectively. So for example the condition∇iZ_k= 0 describes a spacetime in which the stress–energy tensor is constant. WhenT = 0, then the tensorZ= 0, i.e. the spacetimeM could be Ricci flat andM is said to be a vacuum (or empty) (see [20]).

In this paper we introduce a generalization of the condition ∇_iZ_kl = 0 men- tioned above. In this way we will extend the limit of validity of the properties of pseudo Ricci symmetric manifolds using this generalized Einstein gravitational tensor. More precisely, we introduce a new kind of Riemannian manifold whose non-null generalizedZ tensor satisfies the following condition:

∇_kZ_j = 2A_kZ_j+A_jZ_k+AZ_kj. (1.7) Such a manifold is calledpseudoZ symmetric and denoted by (PZS)_n. It is worth to notice that if φ = 0, we recover a pseudo Ricci symmetric manifold, that is, a (PRS)_n manifold, while if Z = g^kZ_k = R+nφ = 0, one has φ = −^R_n and so we recover the classical Z tensor with Z_j =R_j− ^R_ng_j = ⁿ⁻¹_n P_j. Thus the space reduces to apseudoprojective Ricci symmetric manifold, that is, a (PPRS)_n manifold. Hereafter we call the generalizedZ tensor simplyZ tensor.

It is well known that inpseudoRicci symmetric Riemannian manifold the con- ditionA_kR= ¹₂∇kR is true giving a closed 1-formA_k in (1.1). From such a view point and the motivations mentioned above, in our paper we study in more detail the properties ofpseudo Z symmetric manifold focusing our attention to peculiar conditions that give rise to the closeness of the associated 1-formA_k in (1.7).

In particular, we will note that these conditions naturally arise from a (PZS)_n manifold endowed withharmonic curvature tensors: the case with harmonic con- formal curvature tensor will be studied in a special way. Moreover, we will point out how these conditions depend on the choice of the scalar functionφin (1.6).

2. Elementary Properties of a (PZS)_n Manifold

In this section elementary properties of a (PZS)_n are shown. LetM be a non-flat n-dimensional (n≥4) (PZS)_n Riemannian manifold with metricg_ij and Rieman- nian connection∇. We can state the following simple theorem.

Theorem 2.1. Let M be an n-dimensional Riemannian (PZS)_n manifold. If the scalar function φsatisfies the following differential equation:

(∇kφ)g_j=φ(2A_kg_j+A_jg_k+Ag_kj), (2.1) then a(PZS)_n manifold reduces to a(PRS)_n one.

Proof. The proof follows from a straightforward calculation by inserting the defi- nition of the generalizedZ-tensor in Eq. (1.7). IfR_jl= 0, this condition is verified and the manifold is reduced to a trivial pseudo Ricci symmetric one.

Now we point out some useful formulas concerning (PZS)_nmanifolds. Transvect- ing Eq. (1.7) withg^j gives immediately:

∇kZ= 2A_kZ+ 2AZ_k. (2.2)

In the same manner transvecting Eq. (1.7) withg^k and using the relation∇Z_j=

12∇_jR+∇_jφcoming from the contracted second Bianchi identity one obtains

∇kR+ 2∇kφ= 2A_kZ+ 6AZ_k. (2.3) By using both Eqs. (2.2) and (2.3) after a straightforward calculation, one obtains the following results:

AZ_k= 2−n

4 ∇_kφ, (2.4)

and

A_kZ= n−2

4 ∇_kφ+1

2∇_kZ. (2.5)

The last equations are the generalization of the correspondent results given in [4,9].

In fact we can state the following simple remarks.

Remark 2.1. If ∇kφ= 0 with Z = 0 and ∇kR= 0, one hasAZ_k = 0. That is AR_k =−φA_k andA_kZ= ¹₂∇_kZ. SoA_kis a closed 1-form and it is an eigenvector of the Ricci tensor with eigenvalue−φ. In particular ifφ= 0, the results given in [4] are recovered. We have shown that similar results are valid in more general conditions.

Remark 2.2. If Z = 0, then φ = −^R_n. And by a simple calculation we have

∇_kR=∇_kφ= 0. Furthermore, we haveAR_k = ^R_nA_k. So we have obtained that the scalar curvature is a covariant constant and that A_k is an eigenvector of the Ricci tensor with eigenvalue ^R_n. These are the results given in [9].

Remark 2.3. If the scalar curvature is constant, that is,∇kR= 0 withZ= 0 and

∇_kφ= 0, one has∇_kZ =n∇_kφ. Then from Eq. (2.5) it follows immediately that A_kZ= ³ⁿ⁻²_4n ∇kZ. This means that A_k is a closed 1-form.

Remark 2.4. LetM be ann-dimensional Riemannian (PZS)_nmanifold withZ= 0 and ∇kφ = 0 having harmonic curvature tensor which satisfies the property

∇mR^m_jk= 0. ThenA_k becomes a closed 1-form.

Proof. The contracted second Bianchi identity is invoked:

∇mR^m_jk=∇kR_j− ∇jR_k.

Transvecting this with g^k, one obtains∇jR = 0. So we are in the hypothesis of Remark2.3. It is well known that if the space is locally symmetric∇_iR^m_jk= 0 [19]

or Ricci symmetric∇iR_k = 0 [24] and if the Ricci tensor is of Codazzi type, i.e.

∇kR_j=∇jR_k, then one has also ∇mR^m_jk= 0.

Remark 2.5. LetM be ann-dimensional Riemannian (PZS)_n manifold whoseZ- tensor is of Codazzi type, that is,∇_kZ_j=∇_jZ_k. Then: (1)Z_j must be a singular tensor, and (2) if∇φ= 0 we haveZ_j = 0 (a trivial (PZS)_n manifold) that is an Einstein space.

Proof. (1) From (1.7) and the condition ∇_kZ_j = ∇_jZ_k we can easily find A_kZ_j = A_jZ_k. If the Z-tensor is non-singular, i.e. if det(Z_j) = 0, there exists a tensor (Z⁻¹)^s of type (2, 0) with the property (Z_j)(Z⁻¹)^s = δ_j^s. Thus we have

A_kZ_j(Z⁻¹)^s=A_jZ_k(Z⁻¹)^s

and soA_kδ_j^s=A_jδ_k^s. This gives finallyA_k= 0. But the 1-formA_k is supposed to be non-null: thus we must have a singularZ-tensor.

(2) From (2.4) and being ∇_kφ = 0 we have A^lZ_k = 0 and thus A^kA_kZ_j = A_jA^kZ_k = 0 from whichZ_j = 0. This contradicts the definition of a (PZS)_n manifold and so such a kind of manifold can never exist.

Now we consider other curvature tensorsK_jk^m with the usual symmetries of the Riemann tensor satisfying the first Bianchi identity. We can thus state the following theorem.

Theorem 2.2. LetM be ann-dimensional Riemannian manifold having a gener- alized curvature tensorK_jk^m with the property:

∇mK_jk^m =a∇mR^m_jk+b[(∇jR)g_k−(∇kR)g_j], (2.6) where a and b are constants. If ∇mK_jk^m = 0 and the condition b = _2(n−1)^a is satisfied, then the scalar curvatureR is a covariant constant,that is, ∇_kR= 0.

Proof. Transvecting Eq. (2.6) with g^k and using the second contracted Bianchi identity, one easily obtains (∇_jR)[¹₂a−(n−1)b] = 0 from which one concludes immediately.

Corollary 2.1. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0 and ∇kφ = 0 having a generalized curvature tensor which satisfies the property (2.6). If ∇mK_jk^m = 0,thenA_k is a closed 1-form.

Proof. It follows immediately from Remark2.3and Theorem2.2.

Some curvature tensorsK_jk^m with the property (2.6) are well known. We give its examples as follows: the projective curvature tensor P_jk^m [16], the conformal curvature tensorC_jk^m [26,27], the concircular curvature tensor ˜C_jk^m [28], the con- harmonic curvature tensorN_jk^m [24] and the quasi-conformal curvature tensorW_jk^m [29]. So we can state the following corollaries whose proofs are very similar.

Corollary 2.2. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0 and ∇kφ = 0 having harmonic projective curvature tensor, that is, the property ∇_mP_jk^m= 0. Then A_k becomes a closed1-form.

Proof. The components of the projective curvature tensor are defined as (see [16, 24]):

P_jk^m =R^m_jk+ 1

n−1(δ^m_jR_k−δ^m_kR_j). (2.7) Applying the operator of covariant derivative to the previous equation and recalling the second contracted Bianchi identity, one obtains

∇mP_jk^m =n−2

n−1∇mR^m_jk. (2.8)

Thus we are in the condition of Theorem2.2and Corollary2.1.

Corollary 2.3. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0and∇_kφ= 0having harmonic concircular curvature tensor,that is, satisfy- ing the property ∇mC˜_jk^m = 0. Then A_k is a closed 1-form.

Proof. The components of the concircular curvature tensor are defined as (see [26,28]):

C˜_jk^m =R^m_jk+ R

n(n−1)(δ_j^mg_k−δ_k^mg_j). (2.9) Applying the operator of covariant derivative to the previous equation and consid- ering the second contracted Bianchi identity, one obtains

∇_mC˜_jk^m =∇_mR^m_jk+ 1

n(n−1)[(∇_jR)g_k−(∇_kR)g_j]. (2.10) Thus we are in the condition of Theorem2.2and Corollary2.1.

Corollary 2.4. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0 and ∇kφ= 0 having harmonic conharmonic curvature tensor, that is, the property ∇mN_jk^m = 0. Then A_k becomes a closed1-form.

Proof. The components of the conharmonic curvature tensor are defined as (see [24]):

N_jk^m =R^m_jk+ 1

n−2(δ^m_j R_k−δ_k^mR_j+R^m_j g_k−R^m_kg_j). (2.11) Applying the operator of covariant derivative to the previous equation and consid- ering the second contracted Bianchi identity, one obtains

∇_mN_jk^m =n−3

n−2∇_mR^m_jk+ 1

2(n−2)[(∇_jR)g_k−(∇_kR)g_j]. (2.12) Thus we are in the condition of Theorem2.2and Corollary2.1.

Now we focus on Eq. (2.5). The operation of covariant derivation is applied on it and the following relation is obtained:

(∇jA_k)Z+A_k(∇jZ) =n−2

4 ∇j∇kφ+1

2∇j∇kZ. (2.13) Now a similar equation with indicesk and j exchanged is written and then sub- tracted from (2.13) to obtain

(∇jA_k− ∇kA_j)Z+A_k(∇jZ)−A_j(∇kZ) = 0. (2.14) This result will have a discrete consequence in the properties of a (PZS)_n manifold.

Equation (2.5) is substituted in the previous relation and one has immediately:

(∇jA_k− ∇kA_j)Z+2−n

2 [A_k(∇jφ)−A_j(∇kφ)] = 0. (2.15) We can then state the following theorem.

Theorem 2.3. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0 and ∇kφ= 0. Then A_k is a closed 1-form if and only if A_k and∇kφ= 0 are co-directional.

3. The Manifold (PZS)_n with Cyclic Ricci and Z-Tensors

In this section we consider the properties of a (PZS)_n manifold having cyclic Ricci and Z-tensors. An n-dimensional Riemannian manifold is said to be cyclic Ricci tensor if the condition:

∇kR_j+∇jR_k+∇R_kj = 0 (3.1) holds. According to [6], this implies∇kR= 0. So the following theorem also holds.

Theorem 3.1. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z= 0 and∇_kφ= 0 having cyclic Ricci tensor. ThenA_k is a closed 1-form.

Now an analogous definition of cyclicZ-tensor is introduced. Ann-dimensional manifold is said to be cyclicZ-tensor if the following condition holds:

∇_kZ_j+∇_jZ_k+∇Z_kj = 0. (3.2)

The previous equation is now transvected withg^j to give

∇kZ+∇kR+ 2∇kφ= 0. (3.3)

Recalling the relationZ =R+nφthe previous equation can be transformed in the following ones:

∇_kR=−n+ 2

2 ∇_kφ, (3.4)

and

∇_kZ =n−2

2 ∇_kφ. (3.5)

This result is then substituted in Eq. (2.5) to obtain A_kZ= n−2

4 2

n−2∇kZ+1

2∇kZ =∇kZ. (3.6)

We can thus state the following theorem.

Theorem 3.2. Let M be an n-dimensional Riemannian (PZS)_n manifold with Z = 0and∇_kφ= 0 having cyclic Z-tensor. ThenA_k is a closed 1-form.

4. Pseudo Z Symmetric Manifolds with Harmonic Conformal and Quasi-Conformal Curvature Tensors

In this section an n-dimensional (n > 3) (PZS)_n Riemannian manifold with the property ∇_mC_jk^m = 0 and ∇_mW_jk^m = 0 is considered. In other words, we study about a (PZS)_n with harmonic conformal curvature tensor and harmonic quasi- conformal curvature tensor [3]. It is well known that the divergence of the conformal tensor satisfies the relation:

∇_mC_jk^m = n−3 n−2

∇_mR^m_jk+ 1

2(n−1){(∇_jR)g_k−(∇_kR)g_j}

. (4.1) So if we consider∇mC_jkl^m= 0, one immediately obtains

∇_mR_jk^m =∇_kR_j− ∇_jR_k = 1

2(n−1)[(∇_kR)g_j−(∇_jR)g_k]. (4.2) This equation does not match with the hypothesis of Theorem 2.2. So we cannot conclude thatA_k is a closed form in this way. From the contracted second Bianchi identity and from the definition of the Z-tensor Z_k = R_k+φg_k the following equation can be written as

∇mR^m_jk=∇kZ_j− ∇jZ_k+ [(∇jφ)g_k−(∇kφ)g_j]. (4.3) On the other hand, from the definition of a (PZS)_n manifold one easily finds that

∇_kZ_j− ∇_jZ_k=A_kZ_j−A_jZ_k. (4.4)

In this way considering Eqs. (4.2)–(4.4) one can see that the following relation holds for a (PZS)_n manifold withharmonicconformal curvature tensor:

A_kZ_j−A_jZ_k = 1

2(n−1)[∇_k(R+ 2(n−1)φ)g_j

− ∇_j(R+ 2(n−1)φ)g_k]. (4.5) This is the starting point for the proofs of the most important properties of a (PZS)_n manifold havingharmonicconformal curvature tensor, that is,∇mC_jkl^m = 0.

We note that the condition∇_mC_jkl^m = 0 implies that the manifold is a (N CS)_n one: so if the condition∇k(R+ 2(n−1)φ) = 0 is satisfied, theZ-tensor becomes a Codazzi tensor from (4.4) and (4.5). In order to avoid Z_kl to be singular we will suppose ∇_k(R+ 2(n−1)φ) = 0 when we consider Remark 2.5 in the case

∇mC_jkl^m = 0. Moreover, we can state the following remark.

Remark 4.1. A nearly conformal symmetric (PZS)_n Riemannian manifold with R=hφbeing handφconstants can never exist.

Proof. In fact the conditionR=hφ implies∇_k(R+ 2(n−1)φ) = 0. From this, together with (4.5) it follows that

A_kZ_j =A_jZ_k

so the Z-tensor is of Codazzi type and by Remark 2.5 being ∇φ = 0 we achieve Z_j= 0. This contradicts the definition of a (PZS)_n manifold. This result general- izes the previous one obtained in [4].

Now we can state the following fundamental result.

Theorem 4.1. Let M be ann-dimensional (n >3) (PZS)_n Riemannian manifold with the property∇_mC_jk^m = 0. If the tensorZ_k is non-singular,thenA_k is a closed 1-form.

Proof. By performing the covariant derivative of Eq. (4.5) one easily obtains (∇_iA_k)Z_j+A_k(∇_iZ_j)−(∇_iA_j)Z_k−A_j(∇_iZ_k)

= 1

2(n−1)[(∇i∇kρ)g_j−(∇i∇jρ)g_k] (4.6) whereρ=R+ 2(n−1)φdenotes a scalar function. Now a cyclic permutation of the indicesi,j,kis performed and the resulting three equations are added to obtain

(∇iA_k− ∇kA_i)Z_j+ (∇jA_i− ∇iA_j)Z_k

+ (∇_kA_j− ∇_jA_k)Z_i+A_j(∇_kZ_i− ∇_iZ_k)

+A_k(∇iZ_j− ∇jZ_i) +A_i(∇jZ_k− ∇kZ_j) = 0. (4.7)

Inserting Eq. (4.4) in the previous result one easily writes

(∇iA_k− ∇kA_i)Z_j+ (∇jA_i− ∇iA_j)Z_k+ (∇kA_j− ∇jA_k)Z_i= 0. (4.8) Now if the Z-tensor is non-singular, i.e. if det(Z_k)= 0, then there exists a (2,0) tensor (Z⁻¹)^km with the propertyZ_k(Z⁻¹)^km =δ^m . Thus the previous equation is multiplied by (Z⁻¹)^h to obtain

(∇_iA_k− ∇_kA_i)δ^h_j + (∇_jA_i− ∇_iA_j)δ^h_k+ (∇_kA_j− ∇_jA_k)δ_i^h= 0. (4.9) Now we put h=j and sum to obtain

(n−2)(∇iA_k− ∇kA_i) = 0. (4.10) We can thus conclude that ifn >2, thenA_k is a closed 1-form.

Now we consider the case of harmonic quasi-conformal curvature tensor. In 1968 Yano and Sawaki [29] defined and studied a tensorW_jk^m on a Riemannian manifold of dimension n, which includes both the conformal curvature tensor C_jk^m and the concircular curvature tensor ˜C_jk^m as particular cases. This tensor is known asquasi- conformal curvature tensor and its components are given by

W_jk^m =−(n−2)bC_jk^m + [a+ (n−2)b] ˜C_jk^m . (4.11) In the previous equation a = 0, b = 0 are constants and n > 3 since the con- formal curvature tensor vanishes identically for n = 3. A non-flat manifold has the harmonic quasi-conformal curvature tensor if ∇_mW_jk^m = 0. If the equations for ∇mC_jk^m = 0 and∇mC˜_jk^m = 0 are employed and the covariant derivative with respect to the index m is applied on the definition of quasi-conformal curvature tensor, one obtains straightforwardly:

∇_mW_jk^m = (a+b)∇_mR^m_jk

+2a−b(n−1)(n−4)

2n(n−1) [(∇jR)g_k−(∇kR)g_j]. (4.12) Now if∇_mW_jk^m = 0, transvecting the previous equation withg^k after some calcu- lations it follows that

(n−2)a+b(n−2)

n (∇jR) = 0. (4.13) This means that ∇_jR= 0 or a+b(n−2) = 0. Inserting this last result in (4.12), we recover easily Eq. (4.2). If∇jR= 0, Remark2.4is invoked. Then it follows that A_k is a closed 1-form. On the other hand, if Eq. (4.2) is valid, Theorem4.1is used.

Thus we can state the following theorem.

Theorem 4.2. Let M be ann-dimensional(n >3) (PZS)_n Riemannian manifold with the property ∇_mW_jk^m = 0. If the tensor Z_k is non-singular, then A_k is a closed 1-form.

Now we follow the procedure explained in [10] to point out other properties of a (PZS)_n manifold. Transvecting Eq. (4.5) withg^k gives

A_jZ−A^mZ_jm=1

2∇j(R+ 2(n−1)φ). (4.14) Inserting this result in (4.5), one can write the following relation:

A_kZ_j−A_jZ_k = 1

(n−1)[(A_kZ−A^mZ_km)g_j−(A_jZ−A^mZ_jm)g_k]. (4.15) Transvecting the previous equation withA, one straightforwardly obtains

A_kAZ_j=A_jAZ_k. (4.16)

Again we multiply the previous equation byA^j to obtain

A_kA^jAZ_j=A_jA^jAZ_k. (4.17) This last can be rewritten as

AZ_k =A_kA^jAZ_j

A_jA^j =tA_k, (4.18)

wheret= ^Aj_A^AZj

jA^j is a scalar function. We have just proved the following theorem that generalizes a similar result in [10] for an (APRS)_n Riemannian manifold.

Theorem 4.3. Let M be ann-dimensional (n >3) (PZS)_n Riemannian manifold with the property ∇mC_jk^m = 0. Then the vector A is an eigenvector of the Z_k tensor with eigenvaluet.

Inserting (4.18) in Eq. (4.14), one easily obtains A_k(t−Z) =−1

2∇k(R+ 2(n−1)φ). (4.19) This result is again a natural generalization of a similar equation given in [10] for an (APRS)_n Riemannian manifold.

Now transvecting Eq. (4.5) withA^j and using the result (4.18), one straightfor- wardly shows that the following equation holds:

R_k= A_kA A_jA^j

nt−Z n−1

+g_k

Z−t n−1 −φ

. (4.20)

In such a case a Riemannian manifold is said to be quasi-Einstein (see [8]). This result can be written in the more compact form:

R_k =αg_k+βT_kT, (4.21)

where α = ^Z−t_n−1 −φ, β = ^nt−Z_n−1 are the associated scalars and T_k = √^Ak

AjA^j is naturally a unit covector. We have just proved the following theorem.

Theorem 4.4. Let M be ann-dimensional (n >3) (PZS)_n Riemannian manifold with the property∇mC_jk^m = 0. Then M is quasi-Einstein.

Now from Eqs. (4.18) and (2.4) one immediately writes A_kt= 2−n

4 ∇_kφ. (4.22)

The operation of covariant derivation is applied on the previous result and the following relation is obtained:

(∇jA_k)t+A_k∇jt=2−n

4 ∇j∇kφ. (4.23)

Now a similar equation with indices k and j exchanged is written and then sub- tracted from (4.23) to obtain finally:

(∇_jA_k− ∇_kA_j)t=A_j∇_kt−A_k∇_jt. (4.24) We can thus state the following theorem.

Theorem 4.5. Let M be ann-dimensional(n >3) (PZS)_n Riemannian manifold with the property ∇mC_jk^m = 0. Then the relation(4.24)holds.

At this point it is worth to note the following geometric remark.

Remark 4.2. (1) If φ= 0, we recover the (PRS)_n manifold. Thus from (4.22) it follows thatt= 0 and beingZ =R, Eq. (4.20) takes the form:

R_k =−A_kA A_jA^j

R

n−1

+g_k

R

n−1

. (4.25)

So the manifold is quasi-Einstein with opposite associated scalars.

(2) If φ = −^R_n, then Z = 0. Thus we recover the (PPRS)_n manifold. And from Remark2.2we easily obtain that∇kR=∇kφ= 0. So again we havet= 0 and thus finallyR_k =^R_ng_k, that is, the manifold is Einstein.

According to Theorem4.3, a (PZS)_n Riemannian manifold with the property

∇mC_jk^m = 0 is quasi-Einstein, that is, the Ricci tensor satisfiesR_k =αg_k+βT_kT (see [8]). IfZ_k is non-singular, the covectorA_k is closed and by Eq. (4.24) we have A_j(∇kt) = A_k(∇jt). This is taken in conjunction with the equation A_j(∇kZ) = A_k(∇_jZ) coming in the same situation from Eq. (2.14). One easily obtains the following relation:

A_j

∇_knt−Z n−1

=A_k

∇_jnt−Z n−1

. (4.26)

Thus multiplying the previous result by √ ¹

AjA^j and considering Theorem4.3 we can state the following corollary.

Corollary 4.1. LetM be ann-dimensional(n >3) (PZS)_n Riemannian manifold with the property∇_mC_jk^m = 0. Then the manifold M is quasi-Einstein. Moreover, if the tensorZ_k is non-singular,the following holds

T_j(∇_kβ) =T_k(∇_jβ). (4.27)

5. Conformally Flat PseudoZ Symmetric Manifolds: Local Form of the Metric Tensor

In this section we study in depth conformally flat (PZS)_n manifold. In particular we point out the existence of a proper concircular vector in such a manifold and give the local form of the metric tensor. It is worth to notice that the proof in the present paper is based only on the request of a non-singularZ-tensor. First we recall the following theorem whose proof is different from that of [11].

Theorem 5.1. Let M be an n-dimensional (n > 3) manifold whose Ricci tensor is given by R_k =αg_k+βT_kT where T_k is a unit vector. If the manifold is con- formally flat and the conditionT_j(∇_kβ) =T_k(∇_jβ)is satisfied,thenT_k is a proper concircular vector.

Proof. If the manifold is conformally flat, then the following naturally holds:

∇kR_j− ∇jR_k = 1

2(n−1)[(∇kR)g_j−(∇jR)g_k]. (5.1) Equation (4.21) is then substituted in previous relation and the operations of covari- ant differentiation are performed to give straightforwardly:

(∇kβ)T_jT+β(∇kT_j)T+βT_j(∇kT)−(∇jβ)T_kT−β(∇jT_k)T−βT_k(∇jT)

= 1

2(n−1)[(∇kR)g˜ _j−(∇jR)g˜ _k], (5.2) where ˜R=R−2(n−1)α. Recalling thatT_k is a unit vector and so (∇kT)T= 0, Eq. (5.2) is then transvected withg^j to obtain

∇kβ−(∇β)T_kT−β(∇T_k)T−βT_k(∇T) = 1

2∇kR.˜ (5.3) Transvecting again Eq. (5.2) withT^jT gives

∇kβ−(∇β)T_kT−βT(∇T_k) = 1

2(n−1)∇kR˜− 1

2(n−1)(∇R)T˜ _kT. (5.4) Comparing the last two equations gives immediately:

βT_k(∇T) = 2−n

2(n−1)∇kR˜− 1

2(n−1)(∇R)T˜ _kT. (5.5) The last result is then transvected withT^k so that the following holds:

β(∇T) =−1

2(∇R)T˜ . (5.6)

Now Eq. (5.6) is substituted in (5.5) to give

(∇R)T˜ _kT=∇kR.˜ (5.7)